Finite Group Databasebeta

← #13139128

Subgroups — #13139128

Order 1728.

626 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88646C211yesmaximal, characteristic
2M16×C3⋊D9C211yesmaximal, characteristic, complemented
3#11720863C211yesmaximal, characteristic, complemented
4C9×#823C211yesmaximal, characteristic, complemented
5C3×#63383C211yesmaximal, characteristic, complemented
6#11720296C211yesmaximal, characteristic, complemented
7#11720287C211yesmaximal, characteristic, complemented
8#10591001—31nomaximal
9D9×#100—31nomaximal
10C22×D3×D9C412yes—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×#6194C412yes—
15#88646—21no—
16#88646—21no—
17#88646—21no—
18#88646—21no—
19C3×C9×M16C2211yescharacteristic, complemented
20#88532C2211yescharacteristic, complemented
21#88523—21no—
22C8×C3⋊D9—21no—
23C9×#208C2211yescharacteristic, complemented
24C3×#3667C2211yescharacteristic, complemented
25#88414—21no—
26(C3⋊D9)⋊C8—21no—
27C2×#3802—31no—
28M16×C3⋊D3—31no—
29#63818—31no—
30C2×D4×D9—31no—
31C3×#823—31no—
32C3×#823D311yescharacteristic
33#63558—31no—
34#63558—31no—
35M16×D9—31no—
36M16×D9—31no—
37M16×D9—61no—
38C9×#100—31no—
39#63391—31no—
40#63383—31no—
41#63383D311yescharacteristic, complemented
42C22×C3⋊D9C2×C411yescharacteristic
43C22×D3×C9C2×C412yes—
44C22×C3×D9C2×C412yes—
45C2×D3×D9—22no—
46C2×D3×D9—22no—
47C2×D3×D9—44no—
48C2×C3×C4×C9C2311yescharacteristic
49C2×C3⋊Dic9C2×C411yescharacteristic
50C4×C3⋊D9D411yescharacteristic
51C4×C3⋊D9D411yescharacteristic
52C4×C3⋊D9D411yescharacteristic
53C4×C3⋊D9D411yescharacteristic
54C9×D12—21no—
55C9×D12—21no—
56C3×D36—21no—
57C3×D36—21no—
58#6194—22no—
59#6194—22no—
60C3×C8×C9—21no—
61C3⋊(C9⋊C8)—21no—
62C9×C3⋊C8—21no—
63C3×C9⋊C8—21no—
64D3×#100—91no—
65D3×#100—91no—
66C22×D32—32no—
67C2×C4×C3⋊D3—31no—
68C2×C3×D12—31no—
69C2×C3×D12C2×D311yescharacteristic
70C2×#625—32no—
71#3802—61no—
72#3802—61no—
73#3802—61no—
74#3802—61no—
75C23×D9—32no—
76C32×M16C2×D311yescharacteristic
77#3748—31no—
78#3743—61no—
79C8×C3⋊D3—61no—
80C3×#208—31no—
81C3×#208C2×D311yescharacteristic
82#3712—61no—
83#3710—61no—
84C2×D4×C9—31no—
85C2×#611—32no—
86D4×D9—61no—
87D4×D9—61no—
88D4×D9—61no—
89D4×D9—61no—
90C2×D36C2×D311yescharacteristic
91C2×C4×D9—31no—
92C2×C4×D9—61no—
93C9×M16—21no—
94C9×M16—31no—
95C9×M16C2×D311yescharacteristic, complemented
96#3667—31no—
97#3667—61no—
98#3667C2×D311yescharacteristic, complemented
99#3663—61no—
100#3663—61no—
101#3663—62no—
102C8×D9—61no—
103C8×D9—61no—
104C8×D9—62no—
105C22×C3×C9C22×C411yescharacteristic
106C2×C3⋊D9—22no—
107C2×C3⋊D9C22⋊C411yescharacteristic
108C2×C3⋊D9C22⋊C411yescharacteristic
109C2×D3×C9—22no—
110C2×D3×C9—42no—
111C2×C3×D9—22no—
112C2×C3×D9—42no—
113D3×D9—44no—
114D3×D9—44no—
115C3×C4×C9C2×D411yescharacteristic
116C3×C4×C9C2×D411yescharacteristic
117C3⋊Dic9C22⋊C411yescharacteristic
118C3⋊Dic9C22⋊C411yescharacteristic
119C2×D3×D4—91no—
120C2×D3×D4—91no—
121D3×M16—91no—
122D3×M16—91no—
123D3×M16—91no—
124D3×M16—91no—
125D3×M16—181no—
126C3×#100—31no—
127C3×#100—91no—
128#833—91no—
129#833—91no—
130#823—31no—
131#823—91no—
132#823—91no—
133#823D911yescharacteristic, complemented
134C22×C3⋊D3—31no—
135C22×C3×D3—32no—
136C22×C3×D3C4×D312yes—
137C2×D32—62no—
138C2×D32—62no—
139C2×D32—124no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—31no—
142C4×C3⋊D3—31no—
143C4×C3⋊D3—31no—
144C4×C3⋊D3—31no—
145C4×C3⋊D3—31no—
146C3×D12—21no—
147C3×D12—21no—
148C3×D12—61no—
149C3×D12—61no—
150#625—62no—
151#625—62no—
152C23×C9—32no—
153C22×D9—31no—
154C22×D9—61no—
155C22×D9—62no—
156C22×D9—62no—
157C22×D9—124no—
158C22×D9C4×D312yes—
159C32×C8—21no—
160C3⋊(C3⋊C8)—61no—
161C3×C3⋊C8—21no—
162C3×C3⋊C8—61no—
163D4×C9—61no—
164D4×C9—61no—
165C2×C4×C9—21no—
166C2×C4×C9C22×D311yescharacteristic
167#611—62no—
168#611—62no—
169C2×Dic9—31no—
170C2×Dic9—61no—
171D36—21no—
172D36—21no—
173C4×D9—31no—
174C4×D9—31no—
175C4×D9—31no—
176C4×D9—31no—
177C4×D9—61no—
178C4×D9—61no—
179C4×D9—61no—
180C4×D9—61no—
181C8×C9—22no—
182C8×C9—21no—
183C8×C9—61no—
184C9⋊C8—21no—
185C9⋊C8—62no—
186C9⋊C8—61no—
187C2×#100—271noSylow
188C2×C3×C9—21no—
189C2×C3×C9C2×C22⋊C411yescharacteristic
190C3⋊D9—21no—
191C3⋊D9#10012yescomplemented
192D3×C9—42no—
193C3×D9—42no—
194C23×D3—92no—
195C23×D3—92no—
196C2×C3×D4—31no—
197C2×C3×D4—91no—
198C2×#67—92no—
199C2×#67—92no—
200D3×D4—181no—
201D3×D4—181no—
202D3×D4—181no—
203D3×D4—181no—
204D3×D4—181no—
205D3×D4—181no—
206D3×D4—181no—
207D3×D4—181no—
208C2×D12—31no—
209C2×D12C2×D911yescharacteristic
210C2×C4×D3—91no—
211C2×C4×D3—91no—
212C2×C4×D3—181no—
213C3×M16—21no—
214C3×M16—31no—
215C3×M16—91no—
216C3×M16C2×D911yescharacteristic, complemented
217C3×M16D3211yescharacteristic
218#208—31no—
219#208—91no—
220#208—91no—
221#208—181no—
222#208C2×D911yescharacteristic, complemented
223ΓL(1,𝔽25)—181no—
224ΓL(1,𝔽25)—181no—
225ΓL(1,𝔽25)—181no—
226ΓL(1,𝔽25)—181no—
227ΓL(1,𝔽25)—182no—
228D3×C8—181no—
229D3×C8—181no—
230D3×C8—181no—
231D3×C8—181no—
232D3×C8—182no—
233C22×C32C2×C4×D311yescharacteristic
234C2×C3⋊D3—31no—
235C2×C3⋊D3—31no—
236C2×C3⋊D3—62no—
237C2×C3×D3—22no—
238C2×C3×D3—42no—
239C2×C3×D3—62no—
240C2×C3×D3—122no—
241D32—124no—
242D32—124no—
243C32×C4D3×D411yescharacteristic
244C32×C4D3×D411yescharacteristic
245C3⋊Dic3—31no—
246C3⋊Dic3—31no—
247C22×C9—21no—
248C22×C9—62no—
249C22×C9—122no—
250C22×C9C2×C4×D311yescharacteristic
251C2×D9—22no—
252C2×D9—31no—
253C2×D9—31no—
254C2×D9—42no—
255C2×D9—61no—
256C2×D9—61no—
257C2×D9—62no—
258C2×D9—122no—
259C2×D9—124no—
260C2×D9—124no—
261C4×C9—21no—
262C4×C9—21no—
263C4×C9D3×D411yescharacteristic
264C4×C9D3×D411yescharacteristic
265Dic9—31no—
266Dic9—31no—
267Dic9—61no—
268Dic9—61no—
269C22×D4—271no—
270C2×M16—271no—
271C2×M16—271no—
272#100—31no—
273#100—91no—
274#100—271no—
275#100—271no—
276C3×C9C2×#10011yescharacteristic, complemented, Sylow
277C23×C3—32no—
278C23×C3—92no—
279C22×D3—32no—
280C22×D3—91no—
281C22×D3—91no—
282C22×D3—181no—
283C22×D3—182no—
284C22×D3—182no—
285C22×D3—182no—
286C22×D3—182no—
287C22×D3—364no—
288C22×D3—364no—
289C22×D3C4×D912yes—
290C3×D4—61no—
291C3×D4—61no—
292C3×D4—181no—
293C3×D4—181no—
294C2×C3×C4—21no—
295C2×C3×C4C22×D911yescharacteristic
296C2×C3×C4C2×D3211yescharacteristic
297#67—182no—
298#67—182no—
299#67—182no—
300#67—182no—
301C2×Dic3—91no—
302C2×Dic3—91no—
303C2×Dic3—181no—
304D12—21no—
305D12—21no—
306D12—61no—
307D12—61no—
308C4×D3—91no—
309C4×D3—91no—
310C4×D3—91no—
311C4×D3—91no—
312C4×D3—91no—
313C4×D3—91no—
314C4×D3—91no—
315C4×D3—91no—
316C4×D3—181no—
317C4×D3—181no—
318C4×D3—181no—
319C4×D3—181no—
320C3×C8—22no—
321C3×C8—21no—
322C3×C8—21no—
323C3×C8—61no—
324C3×C8—181no—
325C3⋊C8—21no—
326C3⋊C8—61no—
327C3⋊C8—182no—
328C3⋊C8—181no—
329C3⋊C8—181no—
330C2×C32—21no—
331C2×C32D3×C22⋊C411yescharacteristic
332C3⋊D3—32no—
333C3⋊D3—61no—
334C3×D3—42no—
335C3×D3—122no—
336C2×C9—21no—
337C2×C9—21no—
338C2×C9—41no—
339C2×C9—122no—
340C2×C9D3×C22⋊C411yescharacteristic
341D9—32no—
342D9—42no—
343D9—62no—
344D9—61no—
345D9—121no—
346C24—272no—
347C2×D4—31no—
348C2×D4—91no—
349C2×D4—272no—
350C2×D4—541no—
351C2×D4—541no—
352C2×D4—541no—
353C2×D4—541no—
354C22×C4—271no—
355M16—31no—
356M16—91no—
357M16—271no—
358M16—541no—
359M16—541no—
360M16D3×D911yescharacteristic, complemented
361C2×C8—541no—
362C2×C8—541no—
363C22×C3—21no—
364C22×C3—62no—
365C22×C3—122no—
366C22×C3—182no—
367C22×C3—362no—
368C22×C3C2×C4×D911yescharacteristic
369C22×C3C4×D3211yescharacteristic
370C2×D3—22no—
371C2×D3—42no—
372C2×D3—62no—
373C2×D3—91no—
374C2×D3—91no—
375C2×D3—91no—
376C2×D3—91no—
377C2×D3—122no—
378C2×D3—181no—
379C2×D3—181no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—362no—
383C2×D3—364no—
384C2×D3—364no—
385C2×D3—364no—
386C2×D3—364no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4D4×D911yescharacteristic
390C3×C4D4×D911yescharacteristic
391C3×C4#380211yescharacteristic
392C3×C4#381111yescharacteristic
393Dic3—91no—
394Dic3—91no—
395Dic3—91no—
396Dic3—91no—
397Dic3—181no—
398Dic3—181no—
399C32D3×#10011yescharacteristic
400C9—21no—
401C9D3×#10011yescharacteristic, complemented
402C23—32no—
403C23—92no—
404C23—271no—
405C23—542no—
406C23—542no—
407C23—1084no—
408D4—61no—
409D4—61no—
410D4—181no—
411D4—181no—
412D4—542no—
413D4—542no—
414C2×C4—271no—
415C2×C4—271no—
416C2×C4—271no—
417C2×C4—271no—
418C2×C4—271no—
419C2×C4C2×D3×D911yescharacteristic
420C8—21no—
421C8—61no—
422C8—181no—
423C8—541no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—41no—
428C2×C3—122no—
429C2×C3—362no—
430C2×C3C22⋊C4×D911yescharacteristic
431C2×C3#6392111yescharacteristic
432D3—42no—
433D3—92no—
434D3—92no—
435D3—122no—
436D3—182no—
437D3—181no—
438D3—181no—
439D3—361no—
440C22—62no—
441C22—122no—
442C22—182no—
443C22—271no—
444C22—271no—
445C22—362no—
446C22—542no—
447C22—1084no—
448C22—1084no—
449C22C4×D3×D911yescharacteristic
450C4—271no—
451C4—271no—
452C4#8864611yescharacteristic
453C4#8866811yescharacteristic
454C3—21no—
455C3D9×#10011yescharacteristic, complemented
456C3#1059100111yescharacteristic
457C2—21no—
458C2—122no—
459C2—272no—
460C2—362no—
461C2—541no—
462C2#1172104811yescharacteristic
subgroupG/H
centerC2#11721048
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32D3×C22⋊C4
prime radicalC2×#88646C2
cosocleC2×C3×C4×C9C23
hypercenterM16D3×D9
hypocenterC3×C9C2×#100
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. M16×C3⋊D9C2semidirect
3. #11720863C2semidirect
4. C9×#823C2semidirect
5. C3×#63383C2semidirect
6. #11720296C2semidirect
7. #11720287C2semidirect
19. C3×C9×M16C22semidirect
20. #88532C22semidirect
23. C9×#208C22semidirect
24. C3×#3667C22semidirect
41. #63383D3semidirect
95. C9×M16C2×D3semidirect
98. #3667C2×D3semidirect
133. #823D9semidirect
191. C3⋊D9#100semidirect
216. C3×M16C2×D9semidirect
222. #208C2×D9semidirect
276. C3×C9C2×#100semidirect
360. M16D3×D9semidirect
401. C9D3×#100semidirect
455. C3D9×#100semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#886462
M16×C3⋊D92
#117208632
C9×#8232
C3×#633832
#117202962
#117202872
#105910013
D9×#1003